Free monads are useful for many tree-like structures and domain specific languages.

If f is a Functor then the free Monad on f is the type of trees whose nodes are labeled with the constructors of f. The word "free" is used in the sense of "unrestricted" rather than "zero-cost": Free f makes no constraining assumptions beyond those given by f and the definition of Monad. As used here it is a standard term from the mathematical theory of adjoint functors.

Cofree comonads are dual to free monads. They provide convenient ways to talk about branching streams and rose-trees, and can be used to annotate syntax trees. The cofree comonad can be seen as a stream parameterized by a Functor that controls its branching factor.

More information on free monads, including examples, can be found in the following blog posts: https://ekmett.github.io/reader/2008/monads-for-free/ https://ekmett.github.io/reader/2011/free-monads-for-less/

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